For fun I decided to explore this data set a bit to see if I could find something that fits nicely.

I began with the following expression:

 model=a +b (c -d* 2.718281828459045`^(1/(2.` +2.718281828459045`^x x)-1.` E^(e* x^4)))^2 

I then used Mathematica's NonLinearModelFit function to find the values for a, b, c, d, and e.

nlm = NonlinearModelFit[Transpose@data, model, {a, b, c, d, e}, x];

nlm["BestFitParameters"]

Out[301]= {a -> 0.114274, b -> 0.947472, c -> 0.983351, d -> 1.03142, e -> 0.000259904}

This gives a fit that looks like:

It was mentioned in the original post that we expect a limit of around 1.05-1.1, which appears to be captured by this model.

Plotting the residuals we see the following:

Enjoy!

For fun, I decided to explore this data set a bit to see if I could find something that fits nicely.

I began with the following expression:

 model=a +b (c -d* 2.718281828459045`^(1/(2.` +2.718281828459045`^x x)-1.` E^(e* x^4)))^2 

I then used Mathematica's NonLinearModelFit function to find the values for a, b, c, d, and e.

nlm = NonlinearModelFit[Transpose@data, model, {a, b, c, d, e}, x];

nlm["BestFitParameters"]

Out[301]= {a -> 0.114274, b -> 0.947472, c -> 0.983351, d -> 1.03142, e -> 0.000259904}

This gives a fit that looks like:

It was mentioned in the original post that we expect a limit of around 1.05-1.1, which appears to be captured by this model.

Plotting the residuals we see the following:

Enjoy!